Let's analyze the given sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] and determine the truth of each statement.
Firstly, define the sets based on the conditions provided:
- Set [tex]\( A = \{ x \mid x \in U \text{ and } x + 2 > 10 \} \)[/tex]
To find the condition for set [tex]\( A \)[/tex]:
[tex]\[
x + 2 > 10 \implies x > 8
\][/tex]
So, [tex]\( A = \{ x \mid x > 8 \} \)[/tex].
- Set [tex]\( B = \{ x \mid x \in U \text{ and } 2x > 10 \} \)[/tex]
To find the condition for set [tex]\( B \)[/tex]:
[tex]\[
2x > 10 \implies x > 5
\][/tex]
So, [tex]\( B = \{ x \mid x > 5 \} \)[/tex].
Now, let's determine the truth of each pair of statements:
1. [tex]\( 5 \in A ; 5 \in B \)[/tex]
- Check if [tex]\( 5 \in A \)[/tex]:
[tex]\[
5 + 2 > 10 \implies 7 > 10 \quad \text{(False)}
\][/tex]
So, [tex]\( 5 \notin A \)[/tex].
- Check if [tex]\( 5 \in B \)[/tex]:
[tex]\[
2 \cdot 5 > 10 \implies 10 > 10 \quad \text{(False)}
\][/tex]
So, [tex]\( 5 \notin B \)[/tex].
Since both parts are false:
[tex]\[
\text{The statement } 5 \in A ; 5 \in B \text{ is False}.
\][/tex]
2. [tex]\( 6 \in A ; 6 \notin B \)[/tex]
- Check if [tex]\( 6 \in A \)[/tex]:
[tex]\[
6 + 2 > 10 \implies 8 > 10 \quad \text{(False)}
\][/tex]
So, [tex]\( 6 \notin A \)[/tex].
- Check if [tex]\( 6 \in B \)[/tex]:
[tex]\[
2 \cdot 6 > 10 \implies 12 > 10 \quad \text{(True)}
\][/tex]
So, [tex]\( 6 \in B \)[/tex].
Since the first part is false and the second part contradicts by being true:
[tex]\[
\text{The statement } 6 \in A ; 6 \notin B \text{ is False}.
\][/tex]
3. [tex]\( 8 \in A ; 8 \in B \)[/tex]
- Check if [tex]\( 8 \in A \)[/tex]:
[tex]\[
8 + 2 > 10 \implies 10 > 10 \quad \text{(False)}
\][/tex]
So, [tex]\( 8 \notin A \)[/tex].
- Check if [tex]\( 8 \in B \)[/tex]:
[tex]\[
2 \cdot 8 > 10 \implies 16 > 10 \quad \text{(True)}
\][/tex]
So, [tex]\( 8 \in B \)[/tex].
Since the first part is false:
[tex]\[
\text{The statement } 8 \in A ; 8 \in B \text{ is False}.
\][/tex]
4. [tex]\( 9 \in A ; 9 \in B \)[/tex]
- Check if [tex]\( 9 \in A \)[/tex]:
[tex]\[
9 + 2 > 10 \implies 11 > 10 \quad \text{(True)}
\][/tex]
So, [tex]\( 9 \in A \)[/tex].
- Check if [tex]\( 9 \in B \)[/tex]:
[tex]\[
2 \cdot 9 > 10 \implies 18 > 10 \quad \text{(True)}
\][/tex]
So, [tex]\( 9 \in B \)[/tex].
Since both parts are true:
[tex]\[
\text{The statement } 9 \in A ; 9 \in B \text{ is True}.
\][/tex]
So, the correct analysis leads to the conclusion that the correct pair of statements is:
[tex]\[ 9 \in A ; 9 \in B \][/tex]
The final answer is:
[tex]\[ (False, False, False, True) \][/tex]
